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TRANSACTIONS OF THE

AMERICAN MATHEMATICAL SOCIETY

Volume 229, 1977

ALTERNATOR AND ASSOCIATOR IDEAL ALGEBRASBY

IRVIN ROY HENTZEL('), GIULIA MARIA PIACENTINI CATTANEO(2) AND

DENIS FLOYD

Abstract. If / is the ideal generated by all associators, (a, b, c) = (ab)c —

a{bc), it is well known that in any nonassociative algebra R, I C (R, R,

R) + R(R, R, R). We examine nonassociative algebras where / C (R, R,

R). Such algebras include (-1, 1) algebras, Lie algebras, and, as we show, a

large number of associator dependent algebras. An alternator is an associa-

tor of the type (a, a, b), {a, b, a), (f>, a, a). We next study algebras where the

additive span of all alternators is an ideal. These include all algebras where

I = {R, R, R) as well as alternative algebras. The last section deals with

prime, right alternative, alternator ideal algebras satisfying an identity of the

form [x, (x, x, a)] = y(x, x, [x, a]) for fixed y. With two exceptions, if this

algebra has an idempotent e such that (e, e, R) = 0, then the algebra is

alternative. All our work deals with algebras with an identity element over a

field of characteristic prime to 6. All our containment relations are given by

identities.

Introduction. In a nonassociative algebra R, the associator (a, b, c) for a, b,

c elements of R is defined by (a, b, c) = (ab)c - a(bc). Clearly R is

associative if and only if (a, b, c) = 0 for all a, b, c elements of R. If we let I

be the ideal of R generated by all associators, then it is well known that

/ C (R, R, R) + R(R, R, R). We examine algebras over a field F of

characteristic not 2 or 3 which have the property that I Q (R, R, R). We,

however, assume more than the condition that the additive span of associa-

tors is an ideal. We assume that there is an absorption formula of the form

(1) a(b, c,d)= 2 K(ab> c, d)n+ pn(a, be, d)v+ v„(a, b, cd)„.

The constants X^, p„, and vn are elements of F; the subscript m following the

associator means that the unknowns of the expression are to be permuted by the

permutation m. The terms of the summations include all possible ways that

the original four elements can be arranged in an associator. We show that Lie

Received by the editors June 23, 1975.

AMS (MOS) subject classifications (1970). Primary 17A30; Secondary 17E05.Key words and phrases. Associator ideal, right alternative, associator dependent.

(') This work was done while the author held a Science and Humanities Research Institute

grant from Iowa State University.

(2) This work was done while the author was at Iowa State University with a fellowship from

the Consiglio Nazionale delle Ricerche, Italy.

© American Mathematical Society 1977

87

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

88 I. R. HENTZEL, G. M. P. CATTANEO AND DENIS FLOYD

algebras and (-1, 1) algebras are examples of algebras satisfying such a

formula; we show that alternative algebras with an identity element will never

satisfy one (unless they are associative). We call condition (1) the "associator

ideal" condition and the algebras which satisfy it "associator ideal" algebras.

We show that with one exception, any algebra of the (—1, 1) type is an

associator ideal algebra. We also give an example of an algebra in which the

associators form an ideal but which is not an associator ideal algebra.

We extend our concept of associator ideal algebras to alternator ideal

algebras. An alternator is an associator of the form (a, a, b), (a, b, a), (b, a,

a). If we let / be the ideal of a nonassociative algebra R generated by all

alternators, we examine algebras where J is the additive span of alternators.

Again, we assume that the algebra satisfies a specific formula that allows for

the absorption of elements outside alternators into alternators. There are

three types of alternators. Each type of alternator requires a formula for

absorbing multiplication on the right side and on the left side. Any absorp-

tion formula needs six parts. In this.paper we avoid such a general formula,

but to display the generality and complexity, we give the identity for the

absorption of a(b, b, c) G J. The algebra is over a field F of characteristic not

2 and not 3; we can assume a linearized identity. Let M(x,y, z), M'(x,y, z),

and M"(x, y, z) be defined as follows. M(x, y, z) = (x, y, z) + (y, x, z);

M'(x, y, z) = (x, y, z) + (z, y, x); M"(x, y, z) = (x, y, z) + (x, z, y). This

part of the formula can then be written as follows:

aM(b,c,d)= 2 a„M(ab,c,d\+ 2 ß„M'(ab,c,d)nwe S« ir f= S4

+ 2 y„M"(a,bc,d)v+ 2 8,M(a,b,cd\wes4 ies4

+ 2 e„M'(a,bc,d\+ 2 r,„M"(ab,c,d)v.ieS4 ireS4

In the identity aff, /?„, yv, 8„, £„, tl, are elements of the field F; the subscript

permutation m means that the four arguments are to be permuted by m.

We call these algebras "alternator ideal algebras". We show that every

associator ideal algebra with identity is actually an alternator ideal algebra as

well.

A right alternative alternator ideal algebra is a generalization of both (-1,

1) algebras and alternative algebras. If such an algebra R is prime and has an

idempotent e =£ 0, ^ 1 such that (e, e, R) = 0, then with two possible

exceptions the algebra is alternative.

Notation. All of our work assumes that we have an algebra over a field F of

characteristic prime to 6 so that scalar factors of the form \ and {■ are

admissible.


ALTERNATOR AND ASSOCIATOR IDEAL ALGEBRAS 89

As is common in algebra, we often place sets of elements in expressions

where elements should appear. This is to be interpreted as the span of all

things generated when the arguments for the expression are chosen from the

indicated sets. Thus (R, R, R) = the vector space over F spanned by all (a, b,

c) for a,b,cE R.

We use both juxtaposition and "•" to indicate multiplication. In expres-

sions where both appear, the product indicated by juxtaposition is to be taken

first; (a,b,c) = ab- c — a-be.

If R is any algebra, for any a, b G R, we define " » " and "[ ]" by:

a ° b - ab + ba; [a, b] = ab - ba. We indicate (R, +, ° ) by R+ and (R,

+ , [ ]) by R~. We use (a, b, c)+ to mean (a, b, c)+ - (a » b) « c -

a ° (b ° c).

The words "identity" and "formula" refer to equations which hold for all

elements in the ring. The "identity element" is the multiplicative identity 1

such that 1 • a = a • 1 = a for all elements a.

In this paper all containment relations are by formula. We emphasize this

by the symbol Cf. Thus, we write R(R, R, R) Çf(RR, R, R) + (R, RR,R) + (R, R, RR), and we mean that there exists a specific formula of the

form (1) which holds for all elements in R.

If / is an ideal of an algebra R and I2 = 0, we call I a trivial ideal. R is

called semiprime if R has no nonzero trivial ideals. R is called prime if

whenever / and / are ideals such that IJ = JI = 0, then / = 0 or J = 0.

S„ refers to the symmetric group of permutations on zz objects. Permuta-

tions refer to the positions, not the elements. Thus m = (123) acting on a, b, c

is c, a, b. m acting on xx, x2, x3 is x3, xx, x2; m acting on xx, x2, x3 is not xXv,

x2v, x3v = x2, x3, xx. The subscript m which appears in formulas like (1)

means that the arguments are to be permuted by m. For example, if m =

(123)(4) G S4, then by (ab, c, d)v for a, b, c, d E R, we mean (ab, c,

d)„ = (ca,b,d).

The following identity is called the Teichmüller identity. It holds in any

nonassociative algebra and may be verified by expansion.

(2) (ab, c, d) - (a, be, d) + (a, b, cd) = a(b, c, d) + (a, b, c)d.

In a right alternative algebra, the following identities hold; see [7, equations

(2) and (3)].

(3) (a, b, be) = (a, b, c)b, (a, b, b2) = 0, (a, b, c2) - (a, be + cb, c).

Examples of associator ideal algebras. A Lie algebra is a nonassociative

algebra satisfying the identities a2 = 0 and (ab)c + (bc)a + (ca)b = 0. In

such an algebra, (a, b, c) = (ab)c - a(bc) - (ab)c + (bc)a - -(ca)b. Thus

(R, R, R) = R3 is an ideal of R, and it is possible to express the absorption

relation as



x(a, b, c) = -x(ca ■ b) = (ca • b)x = -(b, x, ca).

A nonassociative algebra is called Lie admissible if, under the Lie product

[a, b] = ab - ba, the algebra is a Lie algebra. This is equivalent to

[[>> bl c] + [ib> 4 a] +[[c, a], b] = 0.

This can also be expressed in terms of associators as

(a, b, c) + (b, c, a) + (c, a, b) - (a, c, b) - (c, b, a) - (b, a, c) = 0.

A (- 1, 1) algebra is a right alternative Lie admissible algebra. It is defined

by the identities

(a, b, c) + (a, c, b) = 0, (a, b, c) + (b, c, a) + (c, a, b) = 0.

The absorption formula is given by

x (a, b, c) = - j (xb, c,a)-\ (xa, c,b)-\ (x, b, ca) - j (x, a, cb)

+ j (xa, b,c) + \ (xc, b,a) + \ (x, a, be) + \ (x, c, ba).

Jordan algebras need not be associator ideal algebras. Let Q be the

quaternions; then Q + is a simple Jordan algebra. (Q, Q, Q)+ is spanned by /,

j, and k; hence (Q, Q, Q)+ is not an ideal because it is not all of Q.

We now give an example of an algebra where (R, R, R) is an ideal, but the

algebra is not an associator ideal algebra. A (-1, 1) algebra R is called a

strongly (-1, 1) algebra if [R, [R, R]] = 0. A strongly (-1, 1) algebra is a

(-1, 1) algebra, and so (R, R, R) is an ideal of R. It is immediate that (R, R,

R) is an ideal of R +. In a strongly (-1, 1) algebra (a, b, c)+ = 2(b, a, c); see

[9, equation (14)]. Thus (R, R, R)+ = (R, R, R) is an ideal of R +. We claim

that no formula (1) exists for R +. We refer to the example of a strongly (-1,

1) algebra given in [4, Table I].

If an absorption formula (1) holds in R +, by the correspondence between

associators in R + and associators in R, the right alternative law and the

cyclic law of R, we would have an expression holding of the form:

a o (b, b, a)+ = yx(a, b,a °b) + y2(b, a,a ° b)

+ y3(a,a,b2) + y4(b,b,a2).

Using the properties of strongly (-1, 1) algebras or [4, Table I], we get

2a(b, b, a) = 2exb(a, a, b) + 2e2a(b, b, a) - (ex + e2)(a, b, [a, b]),

where e, = y3 - y, and e2 = y4 - y2. This is a dependence relation among

the bases and we conclude that 2e[ = 0, 2e2 = 2, ex + e2 = 0; this is impossi-

ble.

Associator dependent algebras. Given a nonassociative algebra R over a

field F, R is said to be an associator dependent algebra if it satisfies one or



more identities of the form:

, Yi(fl, b, c) + y2(b, c, a) + y3(c, a, b)

+ y4(a, c, b) + y5(c, b, a) + y6(b, a, c) = 0.

The y,, y2, y3, y4, y5, y6 E F are constants which are not all zero, and the a, b,

c represent any three elements of R. Associator dependent algebras were

classified by Kleinfeld et al. in [8]. Right alternative, left alternative, flexible,

antiflexible, and (-1, 1) algebras are all examples of associator dependent

algebras. Less well known associator dependent algebras have also been

studied.

Let H be the group ring on S3 over F where F is the base field of the

nonassociative algebra. If m E S3, then the subscript m on an associator

means that the arguments of the associator are to permuted by m. We write

permutations on the right; the order of multiplication intended for mo is

shown by (a, b, c)m = [(a, b, c)J0. If A = ~2„eS3y„m, then for a,b,c E R, we

define

(a,b,c)h= 2 Y„(a> ¿> c)„.

Those elements h of H such that (a, b, c)h = 0 for all elements a, b, c of R are

called identities of R (based on the associator). The set of all these identities

of R form a left ideal of H.

The study of the identities of type (4) is equivalent to the study of different

left ideals of H. If we assume that F is not of characteristic 2 or 3, we may

write H = Hx@ H2@ H3, where Hx and H2 are 1 X 1 matrices and H3 is

2x2.

The following representation is used for S3 in this paper.

(12) = (1)©(-1)©(_11 ^J,

(123) = (1)9(1)©(-1 -1).

If L is a left ideal of H, then L = Lx@ L2@ L3. Since Hx and H2 are

one-dimensional vector spaces, their only left ideals are 0 and the whole

space. There are three types of left ideals of H3. They are 0, H3, or a left ideal

generated by a matrix of the form (g $) where a and ß are not both zero. We

shall simply write (g ̂ ) to represent the left ideal of H3 generated by (g g). The

left ideal generated by (g $) corresponds to an identity of this form:

a((a, b, c) + (b, a, c) - (b, c, a) - (c, b, a))

+ ß ((a, c, b) - (c, a, b) + (b, c, a) - (b, a, c)) = 0.



We now list a few well-known structures.

Hx corresponds to third power associativity: a2a = aa2.

H2 corresponds to Lie admissibility.

Hx © H2 corresponds to the cyclic law: (a, b, c) + (b,c, a) + (c, a, b) = 0.

Hx © H3 corresponds to the alternative law: (a, b, c) = (sgn m)(a, b, c)v.

Hx © (¿ ¿) corresponds to the right alternative law: (a, b, c) + (a, c, b) = 0.

Hx® H2@ (I l) corresponds to (— 1, 1) algebras.

An algebra is considered a maximal associator dependent algebra if its left

ideal of identities is a maximal left ideal. One can see that alternative algebras

are certainly maximal. The type described by H2 © H3 is also maximal; this

type is not power associative. They satisfy (a, b, c) = (a, b, c)v for all m E Sy

The third type of maximal associator dependent algebras is defined by

HX®H2® (g g). We shall call them algebras of the (-1, 1) type. (-1, 1)

algebras, flexible Lie admissible algebras, and antiflexible third power

associative algebras are all of this type.

Various types of maximal associator dependent algebras have been studied,

and many results can be found in the literature.

Theorem I. IS R is an associator ideal algebra with an identity element, then

R is Lie admissible.

Proof. We remark that we do not say that R itself is a Lie algebra. We say

that R ~, i.e. R with the product [a, b] = ab — ba, is a Lie algebra. We

assume R satisfies an identity of the type (1).

Consider 2a6S (sgn a)a(b, c, d)a; let d = 1. This sum reduces to

- (a, b, c) - (b, c, a) - (c, a, b) + (a, c, b) + (c, b, a) + (b, a, c)

= - 2 (sgn t)(a, b, c)r.rSS3

If this alternating sum can be shown to be zero, R will be Lie admissible. If

we apply (1) to each of the 24 terms in 2aeS(sgn o)a(b, c, d)„, we have a

sum of 24 • 3 • 24 terms. Fix m E S4. The terms involving A, when collected

become ±\r'2l,eS (sgn o)(ab, c, d)0. Since d = 1, the only nonzero associa-

tors will have d as a factor of the argument in the left entry of the associator.

Each of the associators involving a, b, c will occur twice in the sum because d

will occur as the left factor as well as the right factor of the product entry.

Since these two arrangements will differ by a transposition, they will have

opposite signs and will cancel. Similarly, all pn and vv terms cancel. This

shows that 2TeS(sgn t)(a, b, c)T is an identity of R, and thus R is Lie

admissible.

Corollary. A right alternative associator ideal algebra with an identity

element is a (—1, 1) algebra.



Proof. A (-1, 1) algebra is a right alternative Lie admissible algebra.

Corollary. If R is an alternative associator ideal algebra with an identity

element, then R is associative.

Proof. Since R is Lie admissible, 0 = 2T<SSj(sgn r)(a, b, c)T = 6(a, b, c).

Theorem 2. Except for the cases (£ ¿) and (I ¿), a nonassociative algebra of

the (-1, 1) type is an associator ideal algebra.

Proof. Let K be the group ring on S4 over F. If k = 2w6S4yw7r, associate

with k the formula 2„eS4y„a(6, c, d)„. We let (R)k represent the set of

elements produced by the formula of k as the four arguments vary

throughout R. (R, R, R) may be only an additive group, but the set

N = {k E K\(R)k Qj (R, R, R)) will be a left ideal of the group ring K.We will now show that each associator dependence relation on R generates

two elements of N. If ~2TeS}yT(a, b, c)T = 0 is an associator dependence

relation for R, then we have that

d 2 VT(fl. *. C)T= 0 andt£S}

2 ?v(«,M,T6ÎJ

d = 0

are identities as well. This last identity can be rewritten using (2) as

2 yra(b,c,d\Q}(R,R,R)

where t permutes the first three positions; the fourth position is left fixed.

These are the two elements of N. In our case, the cyclic law and the (g $)

law hold for R; they give us these four elements of N.

kx - (1234) + (14) + (1324),

k2 - / + (123) + (132),

k3 = «((1234) + (134) - (14) - (14)(23))

+ £((124) - (1324) + (14) - (134)),

k4 = a(I + (12) - (132) - (13)) + /?((23) - (123) + (132) - (12)).

As F is not of characteristic 2 or 3, K is isomorphic to Fx x, © Fx x, © F2x2®F3X3 ©^3X3 O.P. 108].

Knowing the images of the generators of SA, (12) and (234) enables one to

compute the entire representation. We include in the listing the other neededelements as well.



G x)(i) © ( i) © (i o\ © /i o o\ © /i o oN

o i o 1 pio0 0 1/ \o o i>

(12) = (1) © (-1) © / i o\ © /-i o o\ © /i o oV-i -i/ -lio l-io

rl 0 1/ \i o -1

(13) (1) © (-1) © /-l -1\ © /l -1 0\ ffi/-l 1 0\ 0 v I 0 -1 o I I 0 1 0

0-1 1/ \0 1 -1

(14) (1) © (-1) © (0 1\ © /l 0 -l\ ®/-l 0 1Vi o/ oi-io-ii

,0 0-1/ \ 0 0 1

(23) = (1) © (-1) © (0 1\ © /0 1 0\ ©/0 -1 0'10 0-100e i)0 0 1/ \0 0 -1

The elements kx, k2, k3, k4 have these representations in the group algebra

(123) = (!)©(!)© /-I -1\ © /-l 1 0\ (n/-l 1 0-10 0-100

-1 0 1/ \-l 0 1

(124) = (1) © ( 1) © / 0 1\ © /-l 0 l\ ©/-l 0 1V-l -1/ -1 1 0 (-1 1 0

V-l 0 0/ \-l 0 0

(132) = (1) © ( 1) © ( 0 1^ © /O -1 0\ © /O -1 01-10 1-10c; -í)

\0 -1 1/ \0 -1 1

(134) ■ (!)©(!)© /-I -IN © /l -1 0\ ffi/l -1 00-1 1 I 0 -1 1

0 -1 0/ \0 -1 0

(234) = (1) © ( 1) © ( 0 1^ © /O 1 0\ © /O 1 0N0 0 1 0 0 1c; -i)

a 0 0/ \1 0 Oy

(1234) = (1) © (-1) © /-I -1\ © /-l 110 1 10-1

■1 0 0/ \1 0 0)

(1324) = (1) © (-1) © / 1 0\ e /O-l l\©/0 1-1V-l -l) 1-10 [-1 1 0

\0 -1 0/ \ 0 1 0

(14) (23) = (1) © ( 1) © /l 0\ 9 /O1 0-1 II 0-1

0 0 -1/ \0 0 -1/



By inspection, one sees that any left ideal containing kx contains both

1 X 1 summands. Similarly, any left ideal containing kx, k2, k3, and kA will

contain both 3x3 summands no matter what values of (a, ß) ^ (0, 0) are

chosen. Any left ideal containing k3 and zc4 will contain the 2 X 2 summand

unless (g $) are of type ($ ¿) or (I ¿). Except for the two exceptional cases, the

left ideal of identities of R(R, R, R) mod (R, R, R) contains all of K. We

have shown (R)I Ç (R, R, R). This means a(b, c, d) C (R, R, R) for all a, b,

c,d E R, and consequently (R, R, R) is an ideal of R.

The exceptional case (° ¿) corresponds to flexible, Lie admissible algebras.

Jordan algebras are of such a type; Q+, where Q is the quaternions, shows

the associators need not be an ideal. The case (q ¿) is the antiflexible, third

power associative algebras. These algebras are specified by the identities (a, b,

c) + (b, c, a) + (c, a, b) = 0 and (a, b, c) = (c, b, a). The structure of these

algebras has been studied in [2], but it is not known if in these rings, the

additive span of the associators is an ideal.

Alternator ideal algebras. We now will briefly discuss alternator ideal

algebras. An alternator is an associator of the form (a, a, b), (a, b, a), or (b, a,

a); i.e. two entries are identical. An alternator ideal algebra is one where the

additive subgroup spanned by the collection of all alternators is an ideal. As

usual, we insist that there be a formula which expresses the absorption. In the

associator ideal case, it was sufficient to find a formula for one-sided

absorption. For alternators, one needs a formula with six parts. We need to

be able to absorb elements on the left side and on the right side for each of

three types of alternators. The simplest examples of alternator ideal algebras

are the alternative algebras, since all alternators of alternative algebras are

zero. The next theorem shows that algebras of type (-1, 1) form another

class of such algebras.

Theorem 3. Any associator ideal algebra with an identity element is an

alternator ideal algebra.

Proof. An associator ideal algebra with an identity element is Lie admissi-

ble by Theorem 1. Thus 0 = 2Tes(sgn j)(a, b, c)T s 6(a, b, c) modulo

alternators. This says that the space spanned by the associators is the same as

the space spanned by the alternators. Since the span of the associators is an

ideal, the span of the alternators is the same ideal.

Right alternative alternator ideal algebras. Let us define M (a, b, c) - (a, b,

c) + (b, a, c). In any algebra satisfying (x, x, x) = 0:

M (a, b,c) = M (b, a, c),

^ M(a,b,c) + M(b,c,a) + M(c,a,b) = 0.



We will examine right alternative algebras satisfying an identity of the

following type.

[a, (b, b, c)] = axM([a, b], b, c) + a2M([a, b], c, b)

(6) +a3M([a,c],b,b)

+ a4M([b, c], a, b) + a5M([b,c], b, a).

By virtue of (5),

M ([R, R], R, R) + M (R, [R, R], R) + M(R,R,[R,R])

CfM([R,R],R,R).

We are thus considering all right alternative algebras where a formula exists

such that

[R,M(R,R,R)] CfM([R,R],R,R)

+ M (R, [R,R],R) + M (R, R, [R, R]).

Theorem 4 (Thedy). Let R be a right alternative algebra. R is an alternator

ideal algebra ** [R, M(R, R, R)] Q} M(R, R, R).

Proof. See [9, Corollary to Lemma 11, p. 23].

Corollary. IS R is a right alternative algebra satisfying (6), then R is an

alternator ideal algebra.

We shall refer to specific algebras satisfying (6) by listing their coefficients

as (a„ a2, a3, a4, a5).

The algebra (0, 0, 0, 0, 0) was given in [9, Theorem 4]. It is clear that [R,

M(R, R, R)] = 0, and, as Thedy showed, M(R, R, R) is an ideal whose

square is zero. It follows that semiprime algebras of type (0, 0, 0, 0, 0) are

alternative. We shall not examine this case any further, and we shall now

assume that in any such formula under discussion, at least one coefficient is

not zero.

The algebra (0, 0, 0, 0, 1) was given in [3] and [5, Equation (5)]. It is a right

alternative algebra satisfying (a, b2, c) = b ° (a, b, c). It is shown in [5] that in

such algebras, M (R, R, R) is an ideal whose square is zero.

A (-1, 1) algebra is of the type (\, - \, 0, - {-, «-).

A strongly (-1, 1) algebra is a (— 1, 1) algebra, so it will satisfy (£, — \,0,

-\, ¿). Since in a strongly (-1, 1) algebra (x, x, [x, a]) = 0, the five terms

on the right-hand side of (6) are dependent. That is:

M ([a, b], b, c) + M([a,b], c,b) + M([a, c],b, b) = 0

and

M ([b, c], a, b) + M([b,c],b,a) + M([a,c], b,b) = 0.



Thus strongly (-1, 1) algebras will satisfy (6) for many choices of coefficients

including (0, - ±, -|, -±,0).

If R is a right alternative alternator ideal algebra, then by hypothesis,

[R,M(R,R,R)] CjM(R,R,R).

This is not (6), because in (6) we assume that

[R, M(R, R, R)] Cf M{[R, R], R,R) + M(R,[R, R], R)

+ M(R,R,[R,R])

CfM([R,R],R,R).

We now ask under what conditions a right alternative alternator ideal algebra

will satisfy (6). If (6) is satisfied, then [a, (a, a, c)] = y(a, a, [a, c]) where

y = - a3- a4 - a5. Such an identity is common. This identity holds in any

flexible algebra for y = 1, and in any algebra of the (-1, 1) type except

flexible Lie admissible for y = 0.

Theorem 5. Let R be a right alternative alternator ideal algebra with an

identity element. R satisfies (6) <=>for some y E F,

[a, (a,a,b)] = y(a,a, [a,b])

is an identity of R.

Proof. We have already shown that (6) implies

[a, (a, a, b)] = (-a3 - a4 - a5)(a,a, [a,b]).

The assumption of alternator ideal and (5) says that

[a,M(b,c,d)]= 2 y„M(ab,c,d)v.l£S4

We can rewrite this as

[a, (b, b, c)] = 8xM(ab,b,c)

+ 82M(ab, c, b) + 83M(ac, b, b)

+ 84M(bc, a, b) + 85M(bc, b, a) + 86M(b2, a, c)

+ 8nM(b\c,a) + M([R,R],R,R)

where the +M([R, R], R, R) is expressed by some formula. In any right

alternative algebra by (2) and (3) we have M(x2, y, x) — 2M(xy, x, x) =

- [x, (x, x, y)] + (x, x, [x, y]). The right-hand side of this identity lies in

M([R, R], R, R) by assumption. If we linearize the left-hand side in three

ways we get the following:



M (b2, a, c) - 2M (ba, b, c) )

+ M (be, a, b) - 2M (ba, c, b) l G M([R, R], R, R)

+ M(cb,a,b) -2M(ca,b,b) )

M (b2, c, a) - 2M (be, b, a) )

+ M(ba,c,b)-2M(bc,a,b) I EM([R, R], R,R)

+ M (ab, c, b) - 2M (ac, b, b) \

M (ab, b, c) - 2M (ab, b, c) "\

+ M (ba, b, c) - 2M (b2, a, c)

+ M(bc, b, a) - 2M(b2, c, a) ( GM([R, R], R, R)

+ M (cb, b, a) - 2M (cb, b, a) (

+ M(ac,b,b) -2M(ab,c,b) \

+ M (ca, b, b) - 2M (cb, a, b) )

These three dependence relations allow us to reduce the number of elements

of the spanning set and to say that

[a, (b, b, c)] = exM(ab, b, c) + e2M(ab,c, b)

+ t3M(ac,b,b) + e4M(bc,a,b) + M([R,R],R,R).

If we linearize this by first replacing a by 1, then c by 1, and finally b by

b + 1, we get

0 = (2ex - e2 - e3)(b, b, c),

0= -(e3 + e4)(b,b,a),

0 = txM(a, b, c) + (e2 + e4)M(c, a, b).

The last equation is an associator dependent equation. Since its representa-

tion in R2x2 is

/ 2e, - (e2 + £4) 0\

^-ei + 2(£2 + e4) 0J'

it will imply R is alternative unless ex = e2 + e4 = 0.

If R is not alternative, then 0 = 2e, - c2 - e3 = £3 + e4 = e, = c2 + £4;

these give £¡ = £2 = £3 = £4 = 0. We have shown that

[a,(b,b,c)] EM([R,R],R,R)

by some formula.

Suppose that A is a right alternative ring satisfying an identity of form (6)

which contains an identity element 1 and an idempotent e ¥= 0,^= 1 such that

(e, e, R) = 0. We will show that with two possible exceptions, if R is prime,

then R is alternative.



From the right alternative property and the assumption on the idempotent,

we can write R as the additive direct sum of the four summands R = Rxx +

Rx0+ Rox + Rm where x G Ry «* ex = ix and xe = jx. This "Peirce" de-

composition has a multiplication between the summands which is given

below; see [6, Lemma 6, p. 166]. We write Rx for Rxx and R0 for Rn*.

Table I

Rx Rx + RQX Rx0 Rx0 0

Rxo 0 *, + *oi *i *io

Rox Rox R0 Rn+ Rx0 0

R0 0 RQX Rox Rl0+ Ro

Furthermore, x20 G Rx and x%x G R0. R* and R0+ are Jordan algebras.

We shall not refer to this table specifically in this paper, but it will be usedrepeatedly in all that we do.

Table I is said to be the table of a right alternative algebra R with an

idempotent e =£ 0,¥= 1 satisfying (e, e, R) = 0. If the products between

summands behave as R0Rk! Q 8jkR¡, except that RXQRX0 ç Rox and R0xR0l C

R\o ($z = I, Sy = 0 if i ¥=j), we shall say that R has an alternative table. In

an alternative table, the summands multiply as matrix units, the exceptions

being RX0RX0 C Rox and ROXROX ç Rxo.

With the possible exception of two cases, these two statements hold. If R is

semiprime, then R has an alternative table. If R is prime, then R is alterna-tive.

We shall use the subscript notation to specify a particular component of a

product we wish to examine. Thus (*10v10)i means the summand of the

product xxoyxo which is in Rx. From the table we see, for example, that

(xx0 yIO)io = 0- Using the table, we often write equalities of the form

[(xioy\o)zioho = (-^-íoJîoîîo-

Since our algebra has an identity element, the idempotent e' = 1 — e

satisfies e' =£ 0,=^ 1 and (e', e', R) = 0. The decomposition of R with respect

to e' is exactly the same except that the subscripts are interchanged. Thus, if

we can show that RX0Rl0 C Rox, by "reversing subscripts" it immediatelyfollows that R0XR0X C Rx0.

In all our proofs, the technique is based on the fact that the identity (6)

holds for all elements in the algebra. We compute the expression for all

possible arrangements of a, b, c, d in [a, (b, c, d) + (c, b, d)]. We then

compare the various equations we get and decide if any (ax, a2, a3, a4, a5) can

possibly satisfy the requirements.

Lemma 1. RXRX ç Rx and R0R0 c R0.



Proof. Since axbx C Rx + RQX, it suffices to show that (axbx)ox = 0. We

know [e, (e, ax, bx) + (ax, e, bx)] = [e, (axbx)ox\ - -(axbx)ox. From (6) we

have

[e, (e, ax, bx) + (ax, e, bx)] G M([e, Rx], R, R) + M([R, R], e, e) = 0.

We have shown (RXRX)0X = 0. The second statement is obtained by reversing

subscripts.

Practically every statement we make will be false for some particular choice

of coefficients (ax, a2, a3, a4, a5). To make the theorems readable, we adopt

the following form. We state the important result of the theorem in the first

sentence. The second sentence gives the exceptions.

We will need to know more about the product A10A10 Q Rx + Rox. In

particular, when does the stronger statement i?10A10 C RQX hold? Suppose

RX0RXQ Z A01. Then there exists an x10 and a y10 such that xxoyXQ = ax + aQX

where ax 7e 0. Let yI0x10 = bx — aox. We use (6) to give us these relations.

M(xxo,yxo,e) M(e,xXQ,yx¿) M(e,yxo,xx¿)

1. [e, M(xxo,yxo, e)] = 2a, - 2a5 a2 ~ «4 «2 - «4

2. [e, M(e, xxo,yx0)] = a2 + a3 + a5 ax a3 + a4

3. [xxo, (e, e,yxo)]= -a2 +a4 -a, a5

4. [xx0, M(e,yx0, e)] = -ax-a3-a4 -a2-a3 -a5

M(xx0,yxo, e) = ax + bx, M(e, xxo,yx0) = - ax, M(e,yxo, xxo) = - bx. These

imply that the left-hand side of each of the four equations listed above is zero.

Interchanging xxo and y10 will give the same equations with ax and bx

interchanged. If we subtract each equation from the one gotten by inter-

changing xlQ and y10, we get (a, - a3 - a4)(ax - bx) = 0, (a, + as)(ax - bx)

= 0 and (a2 + a3 — a5)(ax — bx) = 0. If ax ¥= bx, then these three

coefficients must be 0. Solving them simultaneously implies the algebra is of

the type (a, — a - ß, ß, a — ß, — a). If we substitute these expressions into

the equation, we get a (a, + bx) = 0. This means that ii ax¥1 ± bx, the

algebra must further be restricted to the type (0, - ß, ß, - ß, 0). If ax = bx,

since we assumed ax ̂ 0, the coefficients must satisfy these four equations.

4a, — 2a2 + 2a4 — 4a5 = 0

— a, + 2a2 +a3 —a4 + 2a¡ =0

a, — 2a2 + 2a4 — a5 =0

— 2a, +a2 —a3 — 2a4 +a¡ =0

Solving these simultaneously gives the solution (a, ß, —a - ß, ß, a). By the

above proof and by reversing subscripts, we have shown the following lemma.

Lemma 2. A10A10 C Rox and R0XR0X C Rx0. The only exceptions are the cases

(a, - a - ß, ß, a - ß, - a) and (a, ß, - a - ß, ß, a).



We shall now assume that RXR0X + R0RX0 ¥= 0. We will show that if R is

semiprime, then RXR0X + R0RXo = 0.

Lemma 3. If RXR0X + R0RX0 ¥= 0, íAezz R is of the type (-a, 0, a, 0, 1 - a).

Proof. We may assume that xx yox ¥= 0 for some choice of xx G Rx and

yox E Rox. The other case where RXR0X = 0 but R0RX0 ¥= 0 is done by revers-

ing subscripts.

We examine these equations:

M(xx,yox,e) M(e,yox,xx)

[e, M(e, xx,yox)] = -a3- a5 -a3- a4

[e, M(e,yox,xx)]= -a2 -a,

[e, M(xx,yox, e)] = -ax + a5 -a2 + a4

[xx,(e,e,yQX)]= -a4. -a5

Expanding the alternators gives M(e, xx, v01) = xxyox = - M(xx,yox, e), and

M(e,yox,xx) = 0.We get these equations: a3 + as= 1, a2 = 0, a, - a5 = -1, o4 = 0. The

solution to this system is (-a, 0, a, 0, 1 - a).

Theorem 6. Let R be a right alternative algebra satisfying an identity of form

(6). // R has an identity element 1 and an idempotent e ¥= O,^ 1 such that (e, e,

R) = 0, then RXR0X + RoRl0 is a trivial ideal of R. The case (0, 0, 0, 0, 1) is

excluded.

Proof. We will show (RXR0X)R + R(RxR0l) ç RXR0X + R0RX0. The prod-

ucts involving R0Ri0 are proved by reversing subscripts. We first notice that,

by Lemma 2 and Lemma 3, RxoR\o £ ôi anc* ôiôi C îo- We now

consider the eight cases separately. 1. (RXR0X)RX C RX0RX = 0. 2. (RXR0X)RX0

Q (Rx, Rxo, R0l) C Rx while (RxR0X)Rl0 Q R20 C RQX. Thus (RXR01)RXQ = 0.

3. (RXR0X)R0X is the most difficult and is done last. 4. (RXR0X)R0 C (Rx, R0,

Rox) C RXR0X. 5. RX(RXR0X) Ç (RXRX)R0X + (Rx, Rox, Rx) C RXR0X. 6.

Rl0(RiR0l) £ (*io)2 £ Roi- R¡o(RM £ (*l0. Rou Ri) C Rv Thus

RwiRM = 0*- 1- *oi(*i*oi) £ ô and yet R0X(RXR0X) C (R0XRX)R0X +

(Rox, Rox, Rx) C Rxo. Thus Rol(RxRox) = 0**. 8. R0(RXR0X) C R0RX0. We shall

now prove the omitted case, item 3. [r0l, M(e, xx, xox)] = [rox, xxxox] =

,'oi('xiôi) - (*i*oi)roi " -(îôi)'"« by **■ We have shown

[r0X,M(e,xx,x0X)] = -(xxx0X)rnX E Rx.

Now we expand using the dependence relation and Lemma 3.

[r01, M(e, x„ x01)] - -aM([r0],e], xitxox) - aM([r0l, xx], e, x0l)

aM([r0l, x0]], e, x,) + aM([rox, xoi], xt, e)

(1 - a)M([e, xoi], xx, rox) + (1 - a)A/([x„ x0I], e, r0l)



Since RXORXO Q Rox and A01A0I C Rx0, one has A01 ° A01 = 0***. One can

also show that 0 = M(R0X, e, A01) = M(RX0, e, Rox) = M(RX0, e, Rx) =

M(RX0, Rx, e). Using this we get that

[rox, M(e, xx, xox)] = -aM(rox,xx,x0i) - (1 - a)M(xox, xx, r01).

By (5), this reduces to (2a - l)M(xox, xx, rox) + aM(rox, xox, xx).

M (rQX, xox, xx) = (rox, xox, xx) — (xox> xx, rox)

= ~r0\ ° (x0yxx) + xox(xxrox) = 0

by *** and **. We have now shown that

[rox, M(e, xx, xox)] = (2a - l)M(xox, xx, rox).

Combining these expressions for [rox, M(e, xx, xox)] says

- (*i*oi)roi - (2« - l)#(*oi.*i»'<>i)i- (2a - IX^i^oiKi-

If a =£ 0, then (RxRQX)R0X = 0. This says that RXR0X + R0RX0 is an ideal of R.

By * and ** along with the corresponding results gotten by reversing

subscripts, RXR0X + A0A10 is a trivial ideal.

If a = 0, the algebra is of type (0, 0, 0, 0, 1).

Semiprime. We will show that if R is semiprime, then M(R, R, R) ç Rx +

R0. This will require computing the summands of M(b, c, d) where b, c, d are

chosen from the summands in all 64 possible ways. By Lemma 1 and

Theorem 6, we can improve our multiplication table for the summands.

Theorem 7. Let R be a semiprime, right alternative algebra satisfying an

identity oSSorm (6). //R has an identity element 1 and an idempotent e =£ 0,¥= 1

such that (e, e, R ) = 0, then R has the Sollowing multiplication table Sor the

summands.

Table II

A, AI0 A01 RQ

Rx Rx Rxo 0 0

Rx0 0 Rx + Rox Rx RXQ

Rox Rox R0 Rxo+ RQ 0R0 0 0 A01 A0

If R is semiprime and satisfies (0, 0, 0, 0, 1), we have previously shown [5]

that R is alternative. In this case R satisfies Table II as well.

In this section we shall assume R satisfies the hypotheses of Theorem 7. We

write down all M (b, c, d) where b, c, d are chosen from the summands in all

64 possible ways. The cases M(bx, cx, dx) G A, and M(b0, c0, d0) G A0 are

immediately seen to be in Rx + R0. Of the remaining cases, 38 are zero from

Table II and the right alternative law. Of the 24 remaining cases, since



M (a, b, c) = M(b, a, c),

we really have 16 cases to consider. We examine in detail 8 of the remaining

16; the remainder are gotten by reversing subscripts. In computing the

summands for these 8, we make use of the fact that a20 G Rx and a^x E RQ.

We list the 8 cases that we will study.

1. M(RX, RXQ, Rxo) CRX + Rox 5. M(RX0, Rx0, R0l) = 0

2. M(RX0, Rxo, Rx) C Rx 6. M(RQX, RXQ, Rx0) C Rox + Rx0 + R0

3. M(R0, Rxo, Rxo) CRX + R0l 7. M(RX0, RXQ, Rxo) ç Rx + Rx0

4. M(RXO,RXO,R0)CRX + Rox 8. M(RX0,R0, R0) C Rxo

The next five lemmas will show that all of these eight alternators lie in

Rx + R0. The algebras (- \, 0, \, 0, \) and (\, -\, \, \, - \) may beexceptions. It will be extremely useful to remember that each summand

absorbs multiplication by Rx and R0 and that

M(e,R,R) + M(R,e,R) + M(R,R,e) C Rx + R0.

Since we will be using the various equations again and again, we shall

tabulate them now for easy reference. The elements b, c, d are intended to be

taken from the summands. For x, an element in a summand, it is convenient

to write 8X for the number in { — 1, 0, 1} such that [e, x] = 8xx. In writing

down these equations, we have ignored all alternators containing e as an

argument. Thus, the equations should be interpreted to mean equality modulo

M(e, R, R) + M(R, e, R) + M (R, R, e). We do not claim

M(e,R,R) + M(R,e,R) + M(R,R,e)

is an ideal; it may be only closed under addition.

Table III

M(b, c, d) M{b, d, c)

1. [e, M(b, c, d)] = fyot, + 8c(a¡ - «2) ~Sda3 si>ai -Sca2

2. [c, M(c, d, b)] m 8ba3 +fic(-o, + o2) -$da¡ «¿a, -5ca, +S/-a, + aj

3. [e, M(b, d, ¿)\ = Sba2 -Sda2 Sba¡ -Sca} +«>.,-a2)

4. [b,M(e,c,d)]= -Vi -sdas -V*2 +«>4-«s)

5. [b,M(c,d,e)]= -Sba3 +Sc(-a4+as) +«„«5 -«¡,«3 +5o«5 +«¿(-«4+05)

6. [b,M(e,d,c)]= -Sba2 +fic(a4-a5) -Sba¡ -Sca¡

7. [c, M(e, b, d)] = «c(-o, + aj -Sda4 «ca2 +«¿(-a!4 + aj)

8. [c,M(b,d,e)]= -Sbat +8dat -8^ +8ca, +6¿(a4-a5)

9. [c, M(e, d, b)] s «404 +«c(a, - o2) 8ba5 +8ca,

10. [d,M(e,b,c)]= 8e(-aA + a¡) +8da2 -Scat +8d(-a1 + a2)

11. [d,M(b,c,e)]= -8ba¡ +ôc(a4-a5) +«,,03 -5,,o4 +«ca4

12. [d, M(e, c, b)] = 8ba¡ +8da¡ 8ba4 +8/0,-02)

Lemma 4. M(RX, Rx0, Rxo) c Rx.



Proof. Let us consider ¿>10, cI0, dx and restrict our attention to the A0,

component. We already know M(bl0, cx0, dx)ox = 0. We shall assume

M (bio, dx, cX0)0l¥^ 0.

The appropriate equations are listed below.

M(bx0, dx, cl0)0,

Eq. 3. [e, M(bx0, dx, cxo)]ox = <*i - a3

Eq. 4. [bx0, M(e, cx0, dx)]ox = -a2

Eq. 5. [bx0, M(cx0, dx, e)]0i = -a3 + «5

Eq. 6. [bxo, M(e, dx, cxo)]ox = - a, - a5

Eq. 10. [dx, M(e, bxo, cx0)]ox = -a4

We see that ax — a3 = — 1, - a2 = 0, — a3 + a5 — 0, — a, — a5 = 0, — a4 =

0. The only solution to these five equations is (- £, 0, £,0, |). From Lemma

2 we deduce that A10A10 ç A01 and thus M(bxo, dx, cI0) = - bx0 ° (dxcx¿) =

0. We have shown that, for any choice of ax, a2, a3, a4, as,

M(Rxo,Rx,Rxo)01=0.

We have shown M(Rxo, Rx, AI0) Ç A,; by (5), M(Rx, Rxo> A10) Ç Rx.

Lemma 5. M(R0, Rxo, Rx0) Q Rx; M(RX0, A10, R¿) Q Rx. The only possibleexception is the case (- \, 0, \, 0, \).

Proof. Let us consider bx0, c10, d0 and restrict our attention to the A*0I

component. The appropriate equations are listed here.

M(*io> ci0' ¿o)oi ^(*i0' d0, c10)0I

Eq. 1. [e, M(bxo, c10, dQ)]0X = 2a, - a2 0

Eq. 3. [e, M(bxo, d0, c10)]01 = a2 a, - a3

Eq. 4. [bxo, M(e, cxo, i/0)]01 = - a, - a2

Eq. 5. [¿>I0, M(cI0, d0, e)]ox = - a3 - a4 + a5 -a3 + a5

Eq. 6. [ô10, Af (e, î/0, cI0)]01 s - a2 + a4 - a5 - a, - a5

Eq. 7. [cI0, M(e, 610, i/0)]01 = -ax + a2 a2

Eq. 12. [i/0, M(e, cxo, bX0)]QX = a5 a4

If we assume M(bxo, c10, i/0)01 # 0, then Eq. 1 says 2a, — a2 = —1, and

Eq. 4 + Eq. 7 says -2a, + a2 = 0. This contradiction says that

M(bx0,cx0,d0)Ql= 0.

If M(bxo, d0, c10)oi ¥* 0, from Eq. 3 we get ax- a3= -1, and the remainder

of the listed equations give a2 = 0, — a3 + a5 — 0, — a, - a5 = 0, a4 = 0.

The only solution to these equations is (— \, 0, ^,0, {-).



Lemma 6. M(R0X, Rx0, Rx0) C R0. The only possible exception is the case (j,_ 3 i i _ 1\

4' 4' 4' !>•

Proof. We consider ¿>,0, cox, dx0. These are the appropriate equations.

M(bx0, c01, ¿10)

Eq. 1. [e, M(bx0, cQX, dxo)] s a2- a3

Eq. 4. [è10, M(e, cox, dx0)] = -ax- as

Eq. 5. [bx0, M(cox, dxo, e)] = -a3 + a4

Eq. 6. [6,0, M(e, dxo, cox)] = -a2- a4+ a5

Eq. 7. [c01, M(e, bx0, dx0)] = ax- a2- a4

From Eq. 4, Eq. 5, and Eq. 6 we deduce that M(bx0, cox, dx0) = 0, unless

— a, — a5 — 0, — a3 + a4 = 0, and — a2 — a4 + a5 = 0. This means the

algebra is of the type (a, -a - ß, ß, ß, - a). We now examine the individual

summands of M(bx0, cox, dx¿).

If M(bx0, c0„ í/,o)io 7e 0» then from Eq. 1, a2 - a3 = 1 and from Eq. 7

ax - a2- a4 = 0. The algebra is of the type (0, |, -1, - £, 0). From

Lemma 2 we have A10A10 ç A0, and A01A01 Ç AI0. Thus M(bxo, cox, dxo) —

-(è,0i/,0) « c0, = 0. This is a contradiction, and we must assume

A/(A10,A0„A,0)10=0.

If M(bXQ, cox, dXQ)QX ¥= 0, then from Eq. 1, a2 - a3 = -1. We examine Eq.

7:

[c0„ M(e, bxo, dXQ)] =[c0„ -e(bxodxo)]

= -c0,(e-6,0^,0)= -(c01 -6,0^10)01

— (c0i> bxo, dX0)Ql= Af (610, c01, <^,o)oi-

Eq. 7 says that a, — a2 — a4 — 1. Thus, the algebra is of the type (5, — |, ^,I _1a4' 2/"

We have shown that with one possible exception, M(R0X, Rxo, RXQ) C R0.

Lemma 7. M(RX0, Rxo, Rxo) ç A,.

Proof. Let us consider 610, c10, dx0 and restrict our attention to the A10

component. These are the appropriate equations. We assume that

M(bxo,cxo, dXQ)l0¥= 0.



M(bx0, cxo, dx0)XQ M(bxo, dxo, c10)10

Eq. 1. [e, M(bXQ, c10, dX0)]l0 = 2a, - a2 - a3 0

Eq. 4. [fb10, M(e, cx0, dX0)]XQ = -ax- a5 -a2 + a4 - a5

Eq. 7. [c10, A/(e, bXQ, dxo)]xo = -ax + a2- a4 a2- a4+ a5

Eq. 10. [¿,0, M(e, bxo, c10)]10 = a2- a4+ a5 -ax + a2- a4

Eq. 11. [dx0, M(bx0, cx0, e)]x0 = a3 + a4- 2a5 0

Eq. 12. [dxo, M(e, cxo, bx0)]x0 = ax + a5 a, - a2+ a4

From Eq. 10 + Eq. 11 + Eq. 12 = 0 = (a, + a2 + a3)M(bx0, cx0, dx0)l0,

we have ax + a2 + a3 = 0. From Eq. 1 we have M(bx0, c10, ¿10)10 =

3axM(bX0, cx0, ¿10)10 and thus a, = }. Let us examine Eq. 11.

[dx0, M(bx0, cxo, e)] = -(bxo°cxo)dxo= -M(bx0,cXQ,dx0)x0

= («3 + «4 - 2a5)M(A10, Cio, dxo)xo

and we have a3 + a4 — 2a5 = — 1. Now

[bxo,M(e,cxo,dxo)] =[bx0, -e(cxodxo)] = e(cxodxo) ■ bx0

~ (cio"io' ^io)io= (cio> "io> ̂ io)io= vcio> ̂io> "10)10-

Interchanging the letters b and c gives [cI0, M(e, bx0, dXQ)] = -(bx0, cx0, dXQ)Xn.

From Eq. 4 + Eq. 7 we have

- M(bx0, c,0, ¿io),0= (-2«i + a2- a4- a5)M(bX0, c10, di0)x0

and this implies —2ax + a2 — a4 — a5= -1. Solving these simultaneous

equations: ax + a2 + a3 = 0, ax = 5, a3 + a4 — 2a5 = — 1, and —2a, + a2

- a4 - a5 = -1 gives us that M(Rxo, Rx0, Rx0) C Rx unless the algebra is of

thetype(j,a, -a - \,a, \).

From the proof of Lemma 2 we know that in this case (xx0 y,0), = (yiox\o)i

for all xxo, yxo E Rxo. We conclude (bxocxo-dx0)x0 = (6iocio)i ' ¿10 " (cirAo)i '

dio- (cxobxo • ¿i0)io - by the right alternative law -(c10¿i0- ¿>io)io- Iterating

this three times gives (ô10c10 • </IO)io " _(èir/io' ¿io)io- ̂ ^ (6iocio' ¿10)10 =

0 = (c10610 • ¿io)io> and thus M(A,0, c10, ¿10)10 = 0- This contradicts our

assumption. We have shown that M(RX0, Rxo, Rxo)i0 = 0 and so

M(Rl0, Rxo, Rxo) C Rx.

Lemma 8. M(RX0, R0, R0) = 0.

Proof. We consider A0, c0, ¿10- These are the relevant equations. Notice

that M(RX0, R0, R0) C Rx0; thus, in the equations "=" may be replaced with

equality.

M(bn, ¿iQ, Cn)

Eq. 3. [e, M(b0, ¿10, c0)] = a, - a2

Eq. 10. [¿10, M(e, b0, c0)]= - a, + a2



If M(b0, dx0, c0) 7*= 0, from Eq. 3 we get ax - a2= 1. From Eq. 10 we get

- a, + a2 = 0. This contradiction implies M(A10, A0, A0) = 0.

These last five lemmas give us the following theorem.


identity oSSorm (6). //A has an identity element 1 andan idempotent e =£ 0,=£ 1

such that (e, e, R) = 0, then M(R, R, R) C Rx + A0. The only possible

exceptions are (- {-, 0, \, 0, {) or (\, - f, \, \, - {■).

We intend to show that Ai(A, A, A) and A10A0, + A10 + A01 + A0IA10

are ideals which annihilate each other. This will allow us to show that if A is

prime, then A is alternative. To this end we prove this lemma.

Lemma 9. (A10A10), is an ideal o/A,.

Proof. It is easy to show that (A10A10), is a right ideal of A,. To show it is

a left ideal requires only that [A„ (A,0A,0),] C (RX0RX0)X. Since all

M(e,R,R)l+ M(R,e,R)l+ M(R,R,e)lCe(RxoRxo),

we may continue to use Table III. Calculating modulo e(A10A10), we have

M(bx0, dx, cxo) = — dx • (bxocxo). The relevant equations become:

M(bxo, dx, cx0)

Eq. 2. [e, M(cXQ, dx, bxo)] s -ax + a3

Eq. 4. [bxo> M(e, c,0, dx)] m -a2

Eq. 5. [bxo, M(cxo, dx, e)] = - a3 + a5

Eq. 9. [cxo, M(e, dx, bxo)] = a, + a5

Eq. 10. [dx,M(e,bxo,cxo)]= -a4

Unless - a, + a3 = 0, - a2 = 0, - a3 + a5 = 0, a, + a5 = 0, and — a4 =

1, we must have A, • (A^A^), C (A10A10),. Solving these equations simulta-

neously gives that the algebra is of the type (0, 0, 0, - 1, 0). However, in this

type, by Lemma 2, (A10A10), = 0. Thus it is always true that (A10A10), is an

ideal of A,.


identity of form (6). If R has an identity element 1 and an idempotent e ^ 0,7*= 1

such that (e, e, A) — 0, then R has an alternative table. The only possible

exception is ({■, - f, \, \, - \).

Proof. From Theorem 7 we have all products except A10A10 and A01A01.

When M(R, A, A) Ç A, + A0, it follows that M(R, A, A) • (A,0 + A01) =

(A,o + A01) • M(R, A, A) = 0. Since (A10A10), ç M(R, A, A), we have from

Lemma 9 that (A10A10), is an ideal of A, and it is easy to see it is trivial. We



have proved the theorem except for the possible exceptional case when

M(R, R, R) may not be contained in Rx + R0. In the case (— ¿, 0, \, 0, \)

we know by Lemma 2 that RXnRXQ C Rox and i?0iôi £ Rio* m this case the

theorem also holds. The last remaining case is(^, —\, \, \, —\).

Theorem 10. Let R be aprime, right alternative algebra satisfying an identity

of the form (6). // R has an identity element 1 and an idempotent e =£0,¥= 1

such that (e, e, R ) = 0, then R is alternative. The only possible exceptions are

(-1,0, ¿,0, Candil -f, {-, I -i).

Proof. By Theorem 9, J = RXOROX + îo + ôi + ôiîois an "*eal of R-

By Theorem 8, M(R, R, R) Ç Rx + R0 and M(R, R, R) • (Rx0 + Rox) =

(Rxo + Rox) -M(R,R,R) = 0. It follows that

M(R,R,R)-J = J-M(R,R,R) = 0.

7 = 0 contradicts our assumption that e ^ 0,¥= 1. Thus M(R, R, R) = 0, and

R is alternative.

We proved Theorem 9 and Theorem 10 under the assumption that the ring

had an identity element. We will now show that the assumption of an identity

element is unnecessary. If R is any right alternative algebra over a field F,

thenR# = F x R with operations

(X, r) + (X', r') = (X + X',r + r'), X(X', r') = (XX', Xr'),

(X, r)(X', r') = (XX', Xr' + X'r + rr')

is a right alternative ring. R # contains an isomorphic copy of R and (1, 0) is

the identity element of R #. If R is semiprime, then R # is semiprime. If R

satisfies an identity of form (6), then R # satisfies the same identity. If e is an

idempotent of R, then (0, e) is an idempotent of R#. If we consider

R C R#, then the decomposition of R and R# with respect to an idempo-

tent e E R satisfies Rtj C R #,-,. Theorem 9 implies this corollary.

Corollary. Let R be a semiprime, right alternative algebra satisfying an

identity of form (6). If R has an idempotent e =£0,¥= 1 such that (e, e, R) = 0,

/Aezî R has an alternative table. The only possible exception is (|, — |, ¿, \,

We may drop the hypothesis of an identity element from Theorem 10 as

well. We need to be more careful because if R is prime, R # need not be

prime. The next corollary follows from the proof of Theorem 10 because

when R is prime, either J# n R or M(R #, R #, R #) n R is zero.

Corollary. Let R be a prime, right alternative algebra satisfying an identity

of the form (6). If R has an idempotent e ¥= 0,=^ 1 such that (e, e, R) = 0, then



A is alternative. The only possible exceptions are ( — \, 0, \, 0, \) and (\, —\,

References

1. M. Burrow, Representation theory of finite groups, Academic Press, New York, 1965. MR 38

#250.2. H. Çelik and D. Outcalt, Power-associativity of antiflexible rings (unpublished).

3. S. Getu and D. Rodabaugh, Generalizing alternative rings, Comm. Algebra 2 (1974), 35-81.

MR 50 #4682.4.1. Hentzel, The characterization of(-1,1) rings, J. Algebra 30 (1974), 236-258.5.1. R. Hentzel and G. M. Piacentini Cattaneo, A note on generalizing alternative rings, Proc.

Amer. Math. Soc. 55 (1976), 6-8.6. M. Humm Kleinfeld, On a class of right alternative rings without nilpotent ideals, J. Algebra 5

(1967), 164-174. MR35 #2939.7. E. Kleinfeld, Right alternative rings, Proc. Amer. Math. Soc. 4 (1953), 939-944. MR 15, 595.8. E. Kleinfeld, F. Kosier, J. M. Osborn, and D. Rodabaugh, The structure of associator

dependent rings, Trans. Amer. Math. Soc. 110 (1964), 473-483. MR 28 # 1221.9. A.Thedy, Right alternative rings, J. Algebra 37 (1975), 1-43.

Department of Mathematics, Iowa State University, Ames, Iowa 50011

Department of Mathematics, University of Rome, Rome, Italy

Department of Mathematics, Bogazici Universitesi, Istanbul, Turkey


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